Circle companions of Hardy spaces of the unit disk

TL;DR

This paper characterizes the circle companion of operator-valued Hardy spaces on the unit disk, establishing an isometric isomorphism with a new Hardy space on the circle using SOT measurability.

Contribution

It provides a complete solution to the problem of identifying the circle companion of operator-valued Hardy spaces, introducing a new Hardy space on the circle.

Findings

Constructed a Hardy space on the circle with SOT measurable elements.

Established an isometric isomorphism via a strong Poisson integral.

Solved the classical boundary function problem for operator-valued Hardy spaces.

Abstract

This paper gives a complete answer to the following problem: Find the circle companion of the Hardy space of the unit disk with values in the space of all bounded linear operators between two separable Hilbert spaces. Classically, the problem asks whether for each function $h$ on the unit {\it disk}, there exists a ``boundary function" $bh$ on the unit {\it circle} such that the mapping $bh\mapsto h$ is an isometric isomorphism between Hardy spaces of the unit circle and the unit disk with values in some Banach space. For the case of bounded linear operator-valued functions, we construct a Hardy space of the unit circle such that its elements are SOT measurable, and their norms are integrable: indeed, this new space is isometrically isomorphic to the Hardy space of the unit disk via a ``strong Poisson integral."